76 MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 



calculation of attractions and the mathematical theory of electricity, suppose the co- 

 efficients of every term of the series to be expanded into another series of the sines 

 and cosines of multiple arcs ; and they avail themselves of the property which these 

 terms possess of vanishing, in certain cases, when integrated between certain limits. 

 The success of this plan, however, depends upon the restricting hypothesis above re- 

 ferred to, that the radius vector of the surface of the body is capable of expansion in 

 a series of terms, each of which satisfies Laplace's equation. The following method 

 shows that the coefficient of the general term of the first series is independent of one 

 of the variables, and thus dispenses with the second series of expansions. This re- 

 sult I have arrived at, by first obtaining the integral of Laplace's equation in its 

 most general form, and deducing the arbitrary functions introduced therein, from 

 considerations which enter previous to the equation of the surface of the attracting 

 body. These coefficients being known, it is evident that the attraction of any homo- 

 geneous body on a point within or without it may be immediately found when the 

 equation of its surface is given, since it then depends only on a series of explicit and 

 definite integrations of known functions, which can always be eff^ected, at least ap- 

 proximately. From this, the attraction of a heterogeneous body, similarly circum- 

 stanced, may be found by the usual method of dividing it into concentric layers, and 

 summing the several attractions of these, deduced as above. 



By substituting the attraction so obtained, in the equation of equilibrium of a fluid 

 body, Clairaut's theorem is immediately deduced ; and, from a peculiarity in the 

 functions representing the attraction, it will be seen, that the same principles with 

 longer processes may be carried on indefinitely, without the necessity of actually 

 determining the precise form of those functions. 



The restricted species of spheroid above referred to, comprises all surfaces of revo- 

 lution ; so that it is sufficiently extensive for most practical purposes ; but the inte- 

 gration of Laplace's equation renders the analysis more direct, and the theory more 

 complete. 



On the General Problem of Attractions. 



1. Let f represent the density of a body at the point {x, y, z) ; and let y, g, h be 

 the coordinates of a particle attracted by the body, parallel respectively to the axes 

 x,y, z\ then, if the power of attraction be inversely as the square of the distance, the 

 resolved part of the attraction of the body, parallel to 



. rrr q{f-a;)dxdydz 



7/ i^ rrr q{9-y)dxdydz 



~\~rrr q{h-z)dxdydz 



